2 Answers
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I don't remember any of the prisoner making any comment about Bruce Wayne while he is kept captive in the pit. Therefore, I would think that nobody knows who he is. After all, they are prisoners with few information coming from the outside world. Furthermore, their prison is in a country far away from where Bruce Wayne's activities are famous.

When he escapes, the prisoners cheer up not for who he is, but for what he has done : beaten the pit and showed all the prisoners that it was possible to escape, thus giving them great hope.

They almost certainly didn't know that he was the Batman (nor did they know that some bat-dressing hero even existed).

However, there is this conversation:

Bruce: You told me a child did it.The prisoner: But no ordinary child. A child born in hell. Forged from suffering. Hardened by pain. Not a man from privilege.

So, the prisoner knows that Bruce comes "from privilege". They already know that Bane has a special interest in Bruce, and that he's from Gotham (given his interest in the events there and his references to "my city").

It is not clear if they know his name, but I don't see how would that add any info to what they already knew, since -- being where they were -- it is not likely that they've ever heard of Gotham, let alone Bruce Wayne, before he was put in the pit.